The black sphere model

The integral terms in equation (5.2.7) play a key role in obtaining relation (5.2.1). These terms, despite their limited or even small magnitudes as compared to have a different action radius depending on time. 

In the case of the black sphere model (equation (3.1.1)) describing instant recombination, using the dimensionless variables r = r/ro, t = DI/tq, — 7jnB Cq (see Section 5.1) we can rewrite equation (5.2.5) in the form 

The integration in (5.2.12) is carried out over the surface of a d-dimensional unit radius sphere. 

The reaction radius tq does not enter the asymptotic relation (5.2.1). Its absence at t oo in the spirit of Section 5.1 could be interpreted as emergence of a new spatial scale - the correlation length It arises in (5.2.11) through the terms Wm9m which play the role of the correlation sources. 

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An initial distribution function within geminate pairs directly defines their stability. In terms of the black sphere model, dissimilar defects (v, i) disappear instantly when approaching to within, or when just created by irradiation at the critical relative distance tq (called also clear-cut radius) - see... 

For the black sphere model under consideration the integration in equations (3.2.7) and (3.2.11) over the spherical layer r > ro gives equation (3.2.10) with the flux of particles over the recombination sphere surface... 

In the case of particle interaction described by their drift in potential, the black sphere model again allows us to obtain a simple solution of the kinetics sought for. Imposing the boundary conditions, equation (3.2.17) and w(go, t) = 0, as well as the initial condition (3.2.14) on the motion equation... 

Similarly to the black sphere model, equation (4.1.63), the effective reaction radius i eff could be defined through Kq — 47rDi eff. Comparing the i eff obtained in such a way, with the results of Chapter 3, the conclusion suggests that they coincide, i.e., both definitions of the effective radii turn... 

The conclusion could be drawn from Fig. 4.2 that the steady-state profile depends essentially on the defect mobility or temperature - unlike the black sphere model, equation (4.1.70). The steady-state solution y(r) defines the stationary reaction rate K(00) through the effective radius of reaction R n-... 

For the Coulomb attraction of reactants A and B (eA = — eB) present in equal concentrations n(t) and the black sphere model the kinetic equations read (d = 3)... 

Behaviour of the joint correlation functions (see Figs 6.15 to 6.17 as typical examples of the black sphere model) resembles strongly those demonstrated above for immobile particles at the scale r < = Id the similar particle function exceeds its asymptotic value Xv(r,t) 2> 1. As r Id. both the correlation functions strive for their asymptotics Y(r,t), Xu(r,t) 1. The only peculiarity is that for mobile particles the boundary condition (5.1.40) tends to smooth similar particle correlation near the point r = 0. On the other hand, if one of diffusion coefficients, say Da, is zero, the corresponding... 

Direct establishment of the asymptotic reaction law (2.1.78) requires performance of computer simulations up to certain reaction depths r, equation (5.1.60). In general, it depends on the initial concentrations of reactants. Since both computer simulations and real experiments are limited in time, it is important to clarify which values of the intermediate asymptotic exponents a(t), equation (4.1.68), could indeed be observed for, say, r 3. The relevant results for the black sphere model (3.2.16) obtained in [25, 26] are plotted in Figs 6.21 to 6.23. The illustrative results for the linear approximation are also presented there. 

As it is noted in Section 5.1, a distinctive feature of the linear approximation is the absence of back-coupling between the concentration n(t) and the correlation function Y (r, t) which is also independent of the initial reactant concentrations. Moreover, in the linear approximation the parameter k = D /D does not play any role at all. So, in the black-sphere model for the standard random distribution, Y(r > ro,t) = 1, one gets universal relations (4.1.69), (4.1.65) and (4.1.61) for d = 1,2 and 3 respectively. In contrast, in the superposition approximation the law K — K(t) loses its universality, since along with space dimension d it depends also on both the parameter k and the initial reactant concentrations. 

It was demonstrated in [31] that it is namely the black sphere model which introduces large errors into this accumulation kinetics treated in terms of the superposition approximation. The way of avoiding the superposition approximation s shortcomings was developed in [33] and discussed below. 

A specific feature of the black sphere model is trivial functional discrimination of terms entering the kinetic equations depending if they are related on the defect production or on the spatial correlations. The r.h.s. of equations (7.1.50) to (7.1.52) describe decay of newly-created defect if they find themselves in the recombination volumes of dissimilar defects. If this is not the case, newly created defects can further disappear during diffusive migration. The latter problem was already considered in [14] (see equations (2.1) to (2.3) therein). 

Trapping of particles A by B is described as earlier in terms of the black sphere model (3.2.16). A model of particle reproduction by division (8.2.6) along with a simplification of integral terms has also the following advantage. Creation of particles, as it was shown in Chapter 7 leads usually to the problem of the proper account of free volume available for particles A the superposition approximation is valid here only for small dimensionless particle concentrations. In our treatment of the reproduction this problem does not arise since prey animals A appear near other A s which are outside the... 

The uncorrelated particle distribution (4.1.12) is used, as standard initial conditions for the correlation dynamics. After the transient period the solution (for the stable regime) becomes independent on the initial conditions. For both the joint correlation functions boundary conditions at large distances X (oo, t) = Y(oo, t) = 1 has to be fulfilled due to the correlation weakening. The black sphere model imposes the additional boundary condition (5.1.39) for the correlation function Y(r,t). 

This equation differs from (5.1.4) which served us as an example for calculating the reaction rate in the black sphere model. Introducing the function h(r,t) = a(r)Y(r,t), and taking into account (3.2.16), we arrive at... 

Equation (8.3.14) is not an asymptotically exact result for the black sphere model due to the superposition approximation used. When deriving (8.3.14), we neglected in (8.3.11) small terms containing functionals I[Z], i.e., those terms which came due to Kirkwood s approximation. However, the study of the immobile particle accumulation under permanent source (Chapter 7) has demonstrated that direct use of the superposition approximation does not reproduce the exact expression for the volume fraction covered by the reaction spheres around B s. The error arises due to the incorrect estimate of the order of three-point density p2,i for a large parameter op at some relative distances ( f — f[ < tq, [r 2 - r[ > ro) the superposition approximation is correct, p2,i oc ct 1, however, it gives a wrong order of magnitude fn, oc Oq2 instead of the exact p2,i oc <7q 1 (if n — r[ < ro, fi — f[ < ro). It was... 

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